The closest equivalent of "area" is "obszar". I think area has too many meanings, including for example penalty area (penalty box) in football - so it can be something really small, that couldn't possibly be called region. If English has exactly the very same (in writing) word as Polish, why complicate it? ;)
That's the case, that 'region' is very vague, and 'district' is too specific, too narrowed down. A "district" of a city is "dzielnica".
Polish Wiktionary translates 'precinct' as 'dzielnica' or 'rejon', with the mention that it's e.g. 'rejon komisariatu', so the area on which a given police department has authority
And 'prefecture' is 'prefektura', but my only association with this word is geography of Japan.
While I would agree that the meaning of the two sentences are largely (or even entirely) the same, I feel that in the case of straight-forward sentences such as this it is better form to translate one-to-one rather than paraphrasing. Take for instance 'Oni są w regionie', with just one word changed in Polish you now have to change the entire word-order in English as there is no equivalent agent noun for the verb '(to) be'. That's just my opinion though, and I appreciate your commitment to help Duolingo expand and become more inclusive of acceptable answers, as well as your dedication to daily language studies.